Abstract group 6144.bb: $C_4^4.C_{24}$

• Label: 6144.bb
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{6} \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \)
• Perm deg.: not computed
• Trans deg.: $384$
• Rank: $3$

Group information

Description:	$C_4^4.C_{24}$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_4^2.(C_2^4\times C_{12}).C_2^6.C_2^3$, of order \(1572864\)\(\medspace = 2^{19} \cdot 3 \)
Composition factors:	$C_2$ x 11, $C_3$
Derived length:	$2$

This group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	31	32	480	224	512	768	1024	1024	2048	6144
Conjugacy classes	1	15	2	112	14	128	48	128	64	128	640
Divisions	1	15	1	64	7	32	12	16	8	8	164
Autjugacy classes	1	7	1	16	3	9	4	2	3	1	47

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	384	0	128	0	128	0	0	0	0	640
Irr. rational chars.	8	20	8	20	28	16	32	8	24	164

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$384$
Rank:	$3$
Inequivalent generating triples:	$43680$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{48}=b^{4}=c^{4}=d^{4}=e^{2}=[b,c]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 24 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 8 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_4^2:C_{48})$
Semidirect product:	$(C_4^4.C_8)$ $\,\rtimes\,$ $C_3$	$(C_4^3:C_{16})$ $\,\rtimes\,$ $C_6$	$(C_4^3.C_8)$ $\,\rtimes\,$ $C_{12}$	$(C_4^3:C_6)$ $\,\rtimes\,$ $C_{16}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^4$ . $C_{24}$	$(C_4^4:C_6)$ . $C_4$	$(C_4^4.C_{12})$ . $C_2$	$(C_4^3:C_{24})$ . $C_4$	all 127

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{48} \simeq C_{2} \times C_{4} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}^{2}$
Commutator length:	$1$

Subgroups

There are 32100 subgroups in 7068 conjugacy classes, 405 normal (63 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4\times C_8$	$G/Z \simeq$ $C_4^2:C_6$
Commutator:	$G' \simeq$ $C_4^2$	$G/G' \simeq$ $C_2\times C_4\times C_{48}$
Frattini:	$\Phi \simeq$ $C_2^3\times C_8$	$G/\Phi \simeq$ $C_2^3\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^4.C_8$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_4^4.C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^3:C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^4.C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 6144: $C_4^4.C_{24}$

Order 3072: $C_4^3:C_{48}$ x 4, $C_2^2\times C_4^2:C_{48}$ x 2, $C_4^4.C_{12}$

Order 2048: $C_4^4.C_8$

Order 1536: $C_4^3.C_{24}$ x 12, $C_4^3:C_{24}$ x 4, $C_2^2\times C_4^2:C_{24}$ x 2, $C_2^3:C_{12}\times C_{16}$, $C_4^4:C_6$

Order 1024: $C_4^3:C_{16}$ x 8, $C_2^2\times C_4^2:C_{16}$ x 2, $C_2^2\times C_4^2:C_{16}$ x 2, $(C_2^2\times C_4^2):C_{16}$ x 2, $C_2\times C_4^3\times C_8$

Order 768: $C_4^3.C_{12}$ x 12, $C_4^2:C_{48}$ x 8, $C_4\times A_4\times C_{16}$ x 4, $C_4^3:C_{12}$ x 4, $C_4\wr C_3\times C_2^2$ x 3, $C_2^4:C_{48}$ x 2, $C_4\times C_2^3:C_{24}$

Order 512: $C_4^3.C_8$ x 32, $C_4^3.C_8$ x 24, $(C_2\times C_4^2):C_{16}$ x 16, $C_4^3\times C_8$ x 8, $C_2^2\times C_4^2\times C_8$ x 6, $C_2^4.\OD_{32}$ x 4, $C_2^4.\OD_{32}$ x 4, $C_2^3\times C_4\times C_{16}$ x 2, $C_2\times C_4^4$

Order 384: $C_4^3:C_6$ x 18, $C_2^3:C_{48}$ x 12, $C_4^2:C_{24}$ x 8, $C_4\times C_8\times A_4$ x 4, $C_2^4:C_{24}$ x 2, $C_2\times C_4\times C_{48}$, $C_2\times A_4\times C_4^2$, $C_2^5.A_4$

Order 256: $C_2\times C_4^2\times C_8$ x 96, $C_2^3.\OD_{32}$ x 80, $C_4^2:C_{16}$ x 64, $C_4^2:C_{16}$ x 48, $C_2^3.\OD_{32}$ x 32, $C_2^2\times C_4\times C_{16}$ x 24, $C_2^3\times C_4\times C_8$ x 10, $C_4^4$ x 8, $C_2^2\times C_4^3$ x 7, $C_2^4\times C_{16}$ x 4

Order 192: $C_4\wr C_3$ x 12, $C_2^3:C_{24}$ x 12, $A_4\times C_{16}$ x 8, $C_2^4.A_4$ x 7, $A_4\times C_4^2$ x 4, $C_4\times C_{48}$ x 4, $C_2^4:C_{12}$ x 3, $C_2^2\times C_{48}$ x 2, $C_2\times C_4\times C_{24}$

Order 128: $C_4^2.C_8$ x 256, $C_4^2\times C_8$ x 208, $C_2^2\times C_4\times C_8$ x 200, $C_2\times C_4^3$ x 108, $C_2^3\times C_{16}$ x 56, $C_2\times C_4\times C_{16}$ x 54, $C_2^2.\OD_{32}$ x 32, $C_2^3\times C_4^2$ x 13, $C_2^4\times C_8$ x 4

Order 96: $C_2^3:C_{12}$ x 18, $C_2\times C_{48}$ x 12, $C_8\times A_4$ x 8, $C_4^2:C_6$ x 7, $C_4\times C_{24}$ x 4, $C_2^2\times C_{24}$ x 2, $C_2^3\times A_4$, $C_2\times C_4\times C_{12}$

Order 64: $C_2\times C_4\times C_8$ x 694, $C_2^2\times C_4^2$ x 262, $C_4^3$ x 232, $C_2^2\times C_{16}$ x 156, $C_4:C_{16}$ x 128, $C_2^3\times C_8$ x 72, $C_4\times C_{16}$ x 24, $C_2^4\times C_4$ x 7

Order 48: $C_4\times A_4$ x 12, $C_2\times C_{24}$ x 12, $C_{48}$ x 8, $C_2^2\times A_4$ x 7, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3, $C_4^2:C_3$

Order 32: $C_2\times C_4^2$ x 912, $C_4\times C_8$ x 376, $C_2^2\times C_8$ x 268, $C_2^3\times C_4$ x 138, $C_2\times C_{16}$ x 104, $C_2^5$

Order 24: $C_2\times C_{12}$ x 18, $C_{24}$ x 8, $C_2\times A_4$ x 7, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 542, $C_4^2$ x 494, $C_2\times C_8$ x 216, $C_{16}$ x 16, $C_2^4$ x 15

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7, $A_4$

Order 8: $C_2\times C_4$ x 444, $C_2^3$ x 57, $C_8$ x 32

Order 6: $C_6$ x 7

Order 4: $C_4$ x 64, $C_2^2$ x 57

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6144: $C_4^4.C_{24}$

Order 3072: $C_4^3:C_{48}$, $C_4^4.C_{12}$, $C_2^2\times C_4^2:C_{48}$

Order 2048: $C_4^4.C_8$

Order 1536: $C_2^2\times C_4^2:C_{24}$ x 2, $C_4^3.C_{24}$ x 2, $C_4^3:C_{24}$ x 2, $C_2^3:C_{12}\times C_{16}$, $C_4^4:C_6$

Order 1024: $C_2^2\times C_4^2:C_{16}$ x 2, $C_4^3:C_{16}$ x 2, $C_2\times C_4^3\times C_8$, $C_2^2\times C_4^2:C_{16}$, $(C_2^2\times C_4^2):C_{16}$

Order 768: $C_4^3.C_{12}$ x 5, $C_4\wr C_3\times C_2^2$ x 2, $C_4^3:C_{12}$ x 2, $C_2^4:C_{48}$, $C_4\times C_2^3:C_{24}$, $C_4\times A_4\times C_{16}$, $C_4^2:C_{48}$

Order 512: $C_4^3.C_8$ x 10, $C_2^2\times C_4^2\times C_8$ x 6, $C_4^3.C_8$ x 4, $C_4^3\times C_8$ x 4, $C_2^4.\OD_{32}$ x 2, $C_2^3\times C_4\times C_{16}$ x 2, $(C_2\times C_4^2):C_{16}$ x 2, $C_2^4.\OD_{32}$ x 2, $C_2\times C_4^4$

Order 384: $C_4^3:C_6$ x 6, $C_4^2:C_{24}$ x 3, $C_2^3:C_{48}$ x 2, $C_4\times C_8\times A_4$ x 2, $C_2^4:C_{24}$ x 2, $C_2\times C_4\times C_{48}$, $C_2\times A_4\times C_4^2$, $C_2^5.A_4$

Order 256: $C_2\times C_4^2\times C_8$ x 34, $C_2^3.\OD_{32}$ x 14, $C_2^2\times C_4\times C_{16}$ x 11, $C_2^3\times C_4\times C_8$ x 9, $C_4^2:C_{16}$ x 8, $C_2^3.\OD_{32}$ x 6, $C_2^2\times C_4^3$ x 5, $C_4^4$ x 4, $C_4^2:C_{16}$ x 4, $C_2^4\times C_{16}$ x 2

Order 192: $C_2^3:C_{24}$ x 5, $C_4\wr C_3$ x 4, $C_2^4.A_4$ x 3, $A_4\times C_4^2$ x 2, $C_2^4:C_{12}$ x 2, $C_2^2\times C_{48}$, $C_2\times C_4\times C_{24}$, $A_4\times C_{16}$, $C_4\times C_{48}$

Order 128: $C_2^2\times C_4\times C_8$ x 68, $C_4^2\times C_8$ x 38, $C_2\times C_4^3$ x 31, $C_4^2.C_8$ x 22, $C_2\times C_4\times C_{16}$ x 17, $C_2^3\times C_{16}$ x 13, $C_2^3\times C_4^2$ x 9, $C_2^4\times C_8$ x 4, $C_2^2.\OD_{32}$ x 2

Order 96: $C_2^3:C_{12}$ x 6, $C_8\times A_4$ x 3, $C_4^2:C_6$ x 3, $C_2\times C_{48}$ x 2, $C_4\times C_{24}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^3\times A_4$, $C_2\times C_4\times C_{12}$

Order 64: $C_2\times C_4\times C_8$ x 124, $C_2^2\times C_4^2$ x 66, $C_4^3$ x 38, $C_2^3\times C_8$ x 31, $C_2^2\times C_{16}$ x 25, $C_4:C_{16}$ x 6, $C_2^4\times C_4$ x 5, $C_4\times C_{16}$ x 5

Order 48: $C_2\times C_{24}$ x 5, $C_4\times A_4$ x 4, $C_2^2\times A_4$ x 3, $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_4^2:C_3$, $C_{48}$

Order 32: $C_2\times C_4^2$ x 131, $C_2^2\times C_8$ x 72, $C_4\times C_8$ x 50, $C_2^3\times C_4$ x 41, $C_2\times C_{16}$ x 13, $C_2^5$

Order 24: $C_2\times C_{12}$ x 6, $C_{24}$ x 3, $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 101, $C_4^2$ x 62, $C_2\times C_8$ x 48, $C_2^4$ x 7, $C_{16}$ x 2

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3, $A_4$

Order 8: $C_2\times C_4$ x 78, $C_2^3$ x 19, $C_8$ x 9

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 19, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_4^4.C_{24}$ ($C_1$)

Order 3072: $C_4^3:C_{48}$ ($C_2$) x 4, $C_2^2\times C_4^2:C_{48}$ ($C_2$) x 2, $C_4^4.C_{12}$ ($C_2$)

Order 2048: $C_4^4.C_8$ ($C_3$)

Order 1536: $C_4^3.C_{24}$ ($C_4$) x 8, $C_4^3.C_{24}$ ($C_2^2$) x 4, $C_4^3:C_{24}$ ($C_2^2$) x 2, $C_4^3:C_{24}$ ($C_4$) x 2, $C_2^2\times C_4^2:C_{24}$ ($C_2^2$), $C_2^2\times C_4^2:C_{24}$ ($C_4$), $C_4^4:C_6$ ($C_4$)

Order 1024: $C_4^3:C_{16}$ ($C_6$) x 4, $C_2^2\times C_4^2:C_{16}$ ($C_6$) x 2, $C_2\times C_4^3\times C_8$ ($C_6$)

Order 768: $C_4^2:C_{48}$ ($C_2\times C_4$) x 8, $C_4^3.C_{12}$ ($C_2\times C_4$) x 7, $C_4^3.C_{12}$ ($C_8$) x 4, $C_4\wr C_3\times C_2^2$ ($C_8$) x 2, $C_4^3:C_{12}$ ($C_2\times C_4$) x 2, $C_4^3:C_{12}$ ($C_8$) x 2, $C_4\wr C_3\times C_2^2$ ($C_2\times C_4$), $C_4^3.C_{12}$ ($C_2^3$)

Order 512: $C_4^3.C_8$ ($C_{12}$) x 8, $C_4^3.C_8$ ($C_2\times C_6$) x 4, $C_4^3\times C_8$ ($C_2\times C_6$) x 2, $C_4^3\times C_8$ ($C_{12}$) x 2, $C_2^3\times C_4\times C_{16}$ ($A_4$), $C_2^2\times C_4^2\times C_8$ ($C_2\times C_6$), $C_2^2\times C_4^2\times C_8$ ($C_{12}$), $C_2\times C_4^4$ ($C_{12}$)

Order 384: $C_4^3:C_6$ ($C_{16}$) x 8, $C_4^3:C_6$ ($C_2\times C_8$) x 7, $C_4^2:C_{24}$ ($C_2\times C_8$) x 4, $C_4^2:C_{24}$ ($C_4^2$) x 2, $C_4^2:C_{24}$ ($C_2^2\times C_4$) x 2, $C_4^3:C_6$ ($C_4^2$) x 2, $C_4^3:C_6$ ($C_2^2\times C_4$), $C_2^5.A_4$ ($C_2\times C_8$)

Order 256: $C_4^2:C_{16}$ ($C_2\times C_{12}$) x 8, $C_2\times C_4^2\times C_8$ ($C_2\times C_{12}$) x 7, $C_2^2\times C_4\times C_{16}$ ($C_2\times A_4$) x 4, $C_2\times C_4^2\times C_8$ ($C_{24}$) x 4, $C_4^4$ ($C_2\times C_{12}$) x 2, $C_4^4$ ($C_{24}$) x 2, $C_2^4\times C_{16}$ ($C_2\times A_4$) x 2, $C_2^2\times C_4^3$ ($C_{24}$) x 2, $C_2^3\times C_4\times C_8$ ($C_2\times A_4$), $C_2^2\times C_4^3$ ($C_2\times C_{12}$), $C_2\times C_4^2\times C_8$ ($C_2^2\times C_6$)

Order 192: $C_4\wr C_3$ ($C_2\times C_{16}$) x 8, $C_2^4.A_4$ ($C_2\times C_{16}$) x 4, $C_2^4.A_4$ ($C_4\times C_8$) x 2, $C_4\wr C_3$ ($C_4\times C_8$) x 2, $C_2^4.A_4$ ($C_2^2\times C_8$), $C_4\wr C_3$ ($C_2^2\times C_8$), $C_4\wr C_3$ ($C_2\times C_4^2$)

Order 128: $C_2\times C_4^3$ ($C_{48}$) x 8, $C_2^3\times C_{16}$ ($C_4\times A_4$) x 8, $C_2\times C_4^3$ ($C_2\times C_{24}$) x 7, $C_4^2\times C_8$ ($C_2\times C_{24}$) x 4, $C_2^3\times C_{16}$ ($C_2^2\times A_4$) x 4, $C_2\times C_4^3$ ($C_4\times C_{12}$) x 2, $C_4^2\times C_8$ ($C_2^2\times C_{12}$) x 2, $C_4^2\times C_8$ ($C_4\times C_{12}$) x 2, $C_2^2\times C_4\times C_8$ ($C_2^2\times A_4$) x 2, $C_2^2\times C_4\times C_8$ ($C_4\times A_4$) x 2, $C_2\times C_4^3$ ($C_2^2\times C_{12}$), $C_2^4\times C_8$ ($C_2^2\times A_4$), $C_2^4\times C_8$ ($C_4\times A_4$), $C_2^3\times C_4^2$ ($C_4\times A_4$), $C_2^3\times C_4^2$ ($C_2\times C_{24}$)

Order 96: $C_4^2:C_6$ ($C_4\times C_{16}$) x 4, $C_4^2:C_6$ ($C_2^2\times C_{16}$) x 2, $C_4^2:C_6$ ($C_2\times C_4\times C_8$)

Order 64: $C_4^3$ ($C_2\times C_{48}$) x 8, $C_2^2\times C_{16}$ ($C_2^3:C_{12}$) x 8, $C_2^3\times C_8$ ($C_2^3:C_{12}$) x 7, $C_2^3\times C_8$ ($C_8\times A_4$) x 4, $C_2^2\times C_4^2$ ($C_2\times C_{48}$) x 4, $C_4^3$ ($C_4\times C_{24}$) x 2, $C_2^4\times C_4$ ($C_8\times A_4$) x 2, $C_2^2\times C_4^2$ ($C_8\times A_4$) x 2, $C_2^2\times C_4^2$ ($C_4\times C_{24}$) x 2, $C_2^2\times C_4^2$ ($C_2^3:C_{12}$) x 2, $C_2\times C_4\times C_8$ ($C_4^2:C_6$), $C_4^3$ ($C_2^2\times C_{24}$), $C_4^3$ ($C_2\times C_4\times C_{12}$), $C_2^4\times C_4$ ($C_2^3:C_{12}$), $C_2^3\times C_8$ ($C_2^3\times A_4$), $C_2^2\times C_4^2$ ($C_2^2\times C_{24}$)

Order 48: $C_4^2:C_3$ ($C_2\times C_4\times C_{16}$)

Order 32: $C_2^3\times C_4$ ($A_4\times C_{16}$) x 8, $C_2^3\times C_4$ ($C_2^3:C_{24}$) x 7, $C_2^2\times C_8$ ($C_2^3:C_{24}$) x 4, $C_2\times C_4^2$ ($C_4\times C_{48}$) x 4, $C_2^3\times C_4$ ($A_4\times C_4^2$) x 2, $C_2^2\times C_8$ ($A_4\times C_4^2$) x 2, $C_2^2\times C_8$ ($C_2^4:C_{12}$) x 2, $C_4\times C_8$ ($C_4^2:C_{12}$) x 2, $C_4\times C_8$ ($C_2^4.A_4$) x 2, $C_2\times C_4^2$ ($C_2^2\times C_{48}$) x 2, $C_2^5$ ($C_2^3:C_{24}$), $C_2^3\times C_4$ ($C_2^4:C_{12}$), $C_2^2\times C_8$ ($C_4^2:C_{12}$), $C_2^2\times C_8$ ($C_2^4.A_4$), $C_2\times C_4^2$ ($C_2\times C_4\times C_{24}$), $C_2\times C_4^2$ ($C_4^2:C_{12}$)

Order 16: $C_2^2\times C_4$ ($C_2^3:C_{48}$) x 8, $C_2\times C_8$ ($C_4^2:C_{24}$) x 4, $C_2\times C_8$ ($C_4^3:C_6$) x 4, $C_2^4$ ($C_2^3:C_{48}$) x 4, $C_2\times C_8$ ($C_2\times C_4^2:C_{12}$) x 3, $C_4^2$ ($C_4^2:C_{24}$) x 2, $C_4^2$ ($C_2\times C_4^2:C_{12}$) x 2, $C_2^4$ ($C_4\times C_8\times A_4$) x 2, $C_2^2\times C_4$ ($C_4^2:C_{24}$) x 2, $C_2^2\times C_4$ ($C_4\times C_8\times A_4$) x 2, $C_2\times C_8$ ($C_2^5.A_4$), $C_4^2$ ($C_2\times C_4\times C_{48}$), $C_2^4$ ($C_2^4:C_{24}$), $C_2^2\times C_4$ ($C_2\times C_4^2:C_{12}$), $C_2^2\times C_4$ ($C_2^4:C_{24}$), $C_2^2\times C_4$ ($C_2\times A_4\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_4^2:C_{48}$) x 8, $C_2\times C_4$ ($C_4^3.C_{12}$) x 7, $C_2^3$ ($C_4\times A_4\times C_{16}$) x 4, $C_8$ ($C_4^3.C_{12}$) x 4, $C_2^3$ ($C_2^4:C_{48}$) x 2, $C_2\times C_4$ ($C_2^4.C_2^4.C_3$) x 2, $C_8$ ($C_2^4.C_2^4.C_3$) x 2, $C_8$ ($C_2^4.C_2^4.C_3$) x 2, $C_2^3$ ($C_4\times C_2^3:C_{24}$), $C_2^3$ ($C_4^3.C_{12}$), $C_2\times C_4$ ($C_2^5.C_2^3.C_3$)

Order 4: $C_4$ ($C_4^3.C_{24}$) x 8, $C_2^2$ ($C_4^3.C_{24}$) x 4, $C_2^2$ ($C_4^3:C_{24}$) x 2, $C_4$ ($C_4^3:C_{24}$) x 2, $C_2^2$ ($C_2^3:C_{12}\times C_{16}$), $C_2^2$ ($C_2^2\times C_4^2:C_{24}$), $C_4$ ($C_2^2\times C_4^2:C_{24}$), $C_4$ ($C_2^5.C_2^4.C_3$)

Order 2: $C_2$ ($C_4^3:C_{48}$) x 4, $C_2$ ($C_2^2\times C_4^2:C_{48}$) x 2, $C_2$ ($C_4^4.C_{12}$)

Order 1: $C_1$ ($C_4^4.C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_4^4.C_{24}$ ($C_1$)

Order 3072: $C_4^3:C_{48}$ ($C_2$), $C_4^4.C_{12}$ ($C_2$), $C_2^2\times C_4^2:C_{48}$ ($C_2$)

Order 2048: $C_4^4.C_8$ ($C_3$)

Order 1536: $C_2^2\times C_4^2:C_{24}$ ($C_2^2$), $C_2^2\times C_4^2:C_{24}$ ($C_4$), $C_4^4:C_6$ ($C_4$), $C_4^3.C_{24}$ ($C_2^2$), $C_4^3.C_{24}$ ($C_4$), $C_4^3:C_{24}$ ($C_2^2$), $C_4^3:C_{24}$ ($C_4$)

Order 1024: $C_2\times C_4^3\times C_8$ ($C_6$), $C_2^2\times C_4^2:C_{16}$ ($C_6$), $C_4^3:C_{16}$ ($C_6$)

Order 768: $C_4^3.C_{12}$ ($C_2\times C_4$) x 3, $C_4\wr C_3\times C_2^2$ ($C_2\times C_4$), $C_4\wr C_3\times C_2^2$ ($C_8$), $C_4^3.C_{12}$ ($C_2^3$), $C_4^3.C_{12}$ ($C_8$), $C_4^3:C_{12}$ ($C_2\times C_4$), $C_4^3:C_{12}$ ($C_8$), $C_4^2:C_{48}$ ($C_2\times C_4$)

Order 512: $C_2^3\times C_4\times C_{16}$ ($A_4$), $C_2^2\times C_4^2\times C_8$ ($C_2\times C_6$), $C_2^2\times C_4^2\times C_8$ ($C_{12}$), $C_2\times C_4^4$ ($C_{12}$), $C_4^3.C_8$ ($C_2\times C_6$), $C_4^3.C_8$ ($C_{12}$), $C_4^3\times C_8$ ($C_2\times C_6$), $C_4^3\times C_8$ ($C_{12}$)

Order 384: $C_4^3:C_6$ ($C_2\times C_8$) x 3, $C_4^2:C_{24}$ ($C_2\times C_8$), $C_4^2:C_{24}$ ($C_4^2$), $C_4^2:C_{24}$ ($C_2^2\times C_4$), $C_4^3:C_6$ ($C_4^2$), $C_4^3:C_6$ ($C_2^2\times C_4$), $C_4^3:C_6$ ($C_{16}$), $C_2^5.A_4$ ($C_2\times C_8$)

Order 256: $C_2\times C_4^2\times C_8$ ($C_2\times C_{12}$) x 3, $C_4^4$ ($C_2\times C_{12}$), $C_4^4$ ($C_{24}$), $C_4^2:C_{16}$ ($C_2\times C_{12}$), $C_2^4\times C_{16}$ ($C_2\times A_4$), $C_2^3\times C_4\times C_8$ ($C_2\times A_4$), $C_2^2\times C_4^3$ ($C_2\times C_{12}$), $C_2^2\times C_4^3$ ($C_{24}$), $C_2^2\times C_4\times C_{16}$ ($C_2\times A_4$), $C_2\times C_4^2\times C_8$ ($C_{24}$), $C_2\times C_4^2\times C_8$ ($C_2^2\times C_6$)

Order 192: $C_2^4.A_4$ ($C_2^2\times C_8$), $C_2^4.A_4$ ($C_4\times C_8$), $C_2^4.A_4$ ($C_2\times C_{16}$), $C_4\wr C_3$ ($C_2^2\times C_8$), $C_4\wr C_3$ ($C_4\times C_8$), $C_4\wr C_3$ ($C_2\times C_4^2$), $C_4\wr C_3$ ($C_2\times C_{16}$)

Order 128: $C_2\times C_4^3$ ($C_2\times C_{24}$) x 3, $C_2\times C_4^3$ ($C_2^2\times C_{12}$), $C_2\times C_4^3$ ($C_4\times C_{12}$), $C_2\times C_4^3$ ($C_{48}$), $C_4^2\times C_8$ ($C_2^2\times C_{12}$), $C_4^2\times C_8$ ($C_2\times C_{24}$), $C_4^2\times C_8$ ($C_4\times C_{12}$), $C_2^4\times C_8$ ($C_2^2\times A_4$), $C_2^4\times C_8$ ($C_4\times A_4$), $C_2^3\times C_4^2$ ($C_4\times A_4$), $C_2^3\times C_4^2$ ($C_2\times C_{24}$), $C_2^3\times C_{16}$ ($C_2^2\times A_4$), $C_2^3\times C_{16}$ ($C_4\times A_4$), $C_2^2\times C_4\times C_8$ ($C_2^2\times A_4$), $C_2^2\times C_4\times C_8$ ($C_4\times A_4$)

Order 96: $C_4^2:C_6$ ($C_2\times C_4\times C_8$), $C_4^2:C_6$ ($C_4\times C_{16}$), $C_4^2:C_6$ ($C_2^2\times C_{16}$)

Order 64: $C_2^3\times C_8$ ($C_2^3:C_{12}$) x 3, $C_2\times C_4\times C_8$ ($C_4^2:C_6$), $C_4^3$ ($C_2\times C_{48}$), $C_4^3$ ($C_4\times C_{24}$), $C_4^3$ ($C_2^2\times C_{24}$), $C_4^3$ ($C_2\times C_4\times C_{12}$), $C_2^4\times C_4$ ($C_8\times A_4$), $C_2^4\times C_4$ ($C_2^3:C_{12}$), $C_2^3\times C_8$ ($C_8\times A_4$), $C_2^3\times C_8$ ($C_2^3\times A_4$), $C_2^2\times C_4^2$ ($C_8\times A_4$), $C_2^2\times C_4^2$ ($C_2\times C_{48}$), $C_2^2\times C_4^2$ ($C_4\times C_{24}$), $C_2^2\times C_4^2$ ($C_2^3:C_{12}$), $C_2^2\times C_4^2$ ($C_2^2\times C_{24}$), $C_2^2\times C_{16}$ ($C_2^3:C_{12}$)

Order 48: $C_4^2:C_3$ ($C_2\times C_4\times C_{16}$)

Order 32: $C_2^3\times C_4$ ($C_2^3:C_{24}$) x 3, $C_2^5$ ($C_2^3:C_{24}$), $C_2^3\times C_4$ ($A_4\times C_4^2$), $C_2^3\times C_4$ ($A_4\times C_{16}$), $C_2^3\times C_4$ ($C_2^4:C_{12}$), $C_2^2\times C_8$ ($A_4\times C_4^2$), $C_2^2\times C_8$ ($C_4^2:C_{12}$), $C_2^2\times C_8$ ($C_2^4:C_{12}$), $C_2^2\times C_8$ ($C_2^3:C_{24}$), $C_2^2\times C_8$ ($C_2^4.A_4$), $C_4\times C_8$ ($C_4^2:C_{12}$), $C_4\times C_8$ ($C_2^4.A_4$), $C_2\times C_4^2$ ($C_2^2\times C_{48}$), $C_2\times C_4^2$ ($C_2\times C_4\times C_{24}$), $C_2\times C_4^2$ ($C_4^2:C_{12}$), $C_2\times C_4^2$ ($C_4\times C_{48}$)

Order 16: $C_2\times C_8$ ($C_2\times C_4^2:C_{12}$) x 2, $C_2\times C_8$ ($C_4^2:C_{24}$), $C_2\times C_8$ ($C_4^3:C_6$), $C_2\times C_8$ ($C_2^5.A_4$), $C_4^2$ ($C_4^2:C_{24}$), $C_4^2$ ($C_2\times C_4^2:C_{12}$), $C_4^2$ ($C_2\times C_4\times C_{48}$), $C_2^4$ ($C_2^3:C_{48}$), $C_2^4$ ($C_4\times C_8\times A_4$), $C_2^4$ ($C_2^4:C_{24}$), $C_2^2\times C_4$ ($C_4^2:C_{24}$), $C_2^2\times C_4$ ($C_2^3:C_{48}$), $C_2^2\times C_4$ ($C_4\times C_8\times A_4$), $C_2^2\times C_4$ ($C_2\times C_4^2:C_{12}$), $C_2^2\times C_4$ ($C_2^4:C_{24}$), $C_2^2\times C_4$ ($C_2\times A_4\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_4^3.C_{12}$) x 3, $C_2^3$ ($C_2^4:C_{48}$), $C_2^3$ ($C_4\times C_2^3:C_{24}$), $C_2^3$ ($C_4\times A_4\times C_{16}$), $C_2^3$ ($C_4^3.C_{12}$), $C_2\times C_4$ ($C_2^5.C_2^3.C_3$), $C_2\times C_4$ ($C_2^4.C_2^4.C_3$), $C_2\times C_4$ ($C_4^2:C_{48}$), $C_8$ ($C_2^4.C_2^4.C_3$), $C_8$ ($C_4^3.C_{12}$), $C_8$ ($C_2^4.C_2^4.C_3$)

Order 4: $C_2^2$ ($C_2^3:C_{12}\times C_{16}$), $C_2^2$ ($C_2^2\times C_4^2:C_{24}$), $C_2^2$ ($C_4^3.C_{24}$), $C_2^2$ ($C_4^3:C_{24}$), $C_4$ ($C_2^2\times C_4^2:C_{24}$), $C_4$ ($C_2^5.C_2^4.C_3$), $C_4$ ($C_4^3.C_{24}$), $C_4$ ($C_4^3:C_{24}$)

Order 2: $C_2$ ($C_4^3:C_{48}$), $C_2$ ($C_4^4.C_{12}$), $C_2$ ($C_2^2\times C_4^2:C_{48}$)

Order 1: $C_1$ ($C_4^4.C_{24}$)

Series

Derived series

$C_4^4.C_{24}$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$C_4^4.C_{24}$

$\rhd$

$C_4^3:C_{48}$

$\rhd$

$C_4^3.C_{24}$

$\rhd$

$C_4^3.C_{12}$

$\rhd$

$C_2\times C_4^2\times C_8$

$\rhd$

$C_4^2\times C_8$

$\rhd$

$C_4^3$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4^4.C_{24}$

$\rhd$

$C_4^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4\times C_8$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $640 \times 640$ character table is not available for this group.

Rational character table

See the $164 \times 164$ rational character table (warning: may be slow to load).